Hecke algebras of finite type are cellular

Let \cH be the one-parameter Hecke algebra associated to a finite Weyl group $W$, defined over a ground ring in which ``bad'' primes for $W$ are invertible. Using deep properties of the Kazhdan--Lusztig basis of \cH and Lusztig's \ba-function, we show that \cH has a natural cellular structure in the sense of Graham and Lehrer. Thus, we obtain a general theory of ``Specht modules'' for Hecke algebras of finite type. Previously, a general cellular structure was only known to exist in types $A_n$ and $B_n$.Comment: 14 pages; added reference

Eigenvalues of real symmetric matrices

We present a proof of the existence of real eigenvalues of real symmetric matrices which does not rely on any limit or compactness arguments, but only uses the notions of "sup", "inf".Comment: 2 pages; appears in the Amer. Math. Monthly (2015

Computing Kazhdan--Lusztig cells for unequal parameters

Following Lusztig, we consider a Coxeter group $W$ together with a weight function $L$. This gives rise to the pre-order relation $\leq_{L}$ and the corresponding partition of $W$ into left cells. We introduce an equivalence relation on weight functions such that, in particular, $\leq_{L}$ is constant on equivalent classes. We shall work this out explicitly for $W$ of type $F_4$ and check that several of Lusztig's conjectures concerning left cells with unequal parameters hold in this case, even for those parameters which do not admit a geometric interpretation. The proofs involve some explicit computations using {\sf CHEVIE}